A wide variety of different drinking straw covers and other types of drinking straw wrappers are known in the art. A typical conventional arrangement is described in U.S. Pat. No. 5,054,264, issued Oct. 8, 1991 and entitled “Easy Tear Straw Cover.” This drinking straw cover includes a line of circumferential indentations that separate the straw cover into an upper proximal portion and a remaining distal portion. The circumferential indentations are configured to facilitate separation of the straw cover into its upper proximal and remaining distal portions when the straw cover is opened by a user pulling it apart along its longitudinal axis.